Stochastic Reduced Basis Methods
Stochastic Reduced Basis Methods
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DOI：
10.2514/2.1837
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发表时间：
2002
期刊：
影响因子：
2.5
通讯作者：
P. Nair;A. Keane
中科院分区：
文献类型：
--
作者：
P. Nair;A. Keane
介绍了求解大规模线性随机代数方程组的随机约简基方法，如将线性随机偏微分方程组在空间、时间和随机维度上离散得到的方法。所采用的基本思想是使用具有未定确定性系数(或随机函数)的随机基向量的线性组合来表示系统响应。我们给出了使用预条件随机Krylov子空间中的基向量来逼近响应过程的理论证明。随后，采用Bubnov-Galerkin格式的变体来计算待定系数，从而得到响应量的显式表达式。我们还研究了投影方案的一些理论性质和计算响应统计量的过程。对随机结构体系的静力和动力分析进行了数值研究。我们证明，与诺伊曼展开方案以及文献中的其他相关技术相比，可以实现显著的改进。
Stochastic reduced basis methods for solving large-scale linear random algebraic systems of equations, such as those obtained by discretizing linear stochastic partial differential equations in space, time, and the random dimension, are introduced. The fundamental idea employed is to represent the system response using a linear combination of stochastic basis vectors with undetermined deterministic coefficients (or random functions). We present a theoretical justification for employing basis vectors spanning the preconditioned stochastic Krylov subspace to approximate the response process. Subsequently, variants of the Bubnov–Galerkin scheme are employed to compute the undetermined coefficients, which allow explicit expressions for the response quantities to be derived. We also examine some theoretical properties of the projection scheme and procedures for computing the response statistics. Numerical studies are presented for static and dynamic analysis of stochastic structural systems. We demonstrate that significant improvements over the Neumann expansion scheme, as well as other relevant techniques in the literature, can be achieved.